The 3B+ is the same price. It is the same price as the 3B but it might be at a slight premium due to it being new. If buying get a 3B+ it is better on several fronts including faster WiFi, ~17% faster CPU, and power over Ethernet.

Just get a 3B+ as GeForce now uses a lot of bandwidth. The 3B+ has noticeably faster WiFi and Ethernet. Video game streaming needs as real good Networking as possible to work effectively. Not to mention Pi s in general need all the speed they can get. Just a fair word of warning, just because a place has internet does not mean it is fast enough for remote gaming. If your household internet is fast enough go ahead and make a project out of that. It sounds like a great idea.